Download this article
 Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Contact three-manifolds with exactly two simple Reeb orbits

Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui Liu

Geometry & Topology 27 (2023) 3801–3831
Abstract

It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

Keywords
Reeb orbit, embedded contact homology, lens space, pseudorotation
Mathematical Subject Classification
Primary: 37J99, 53E50
Secondary: 53D42
References
Publication
Received: 29 September 2021
Revised: 24 January 2022
Accepted: 26 February 2022
Published: 5 December 2023
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Leonid Polterovich
Authors
Daniel Cristofaro-Gardiner
Department of Mathematics
University of California, Santa Cruz
Santa Cruz, CA
United States
School of Mathematics
Institute for Advanced Study
Princeton, NJ
United States
Department of Mathematics
University of Maryland
College Park, MD
United States
Umberto Hryniewicz
RWTH Aachen
Aachen
Germany
Michael Hutchings
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Hui Liu
School of Mathematics and Statistics
Wuhan University
Wuhan
China

Open Access made possible by participating institutions via Subscribe to Open.